Theorems · Theorem · global analysis

Manifold.IsSubmersionAt.prodMap

∀ {𝕜 : Type u_1} {E' : Type u_2} {E'' : Type u_3} {E''' : Type u_4} {H : Type u_7} {H' : Type u_8} {G : Type u_9}
  {G' : Type u_10} {E : Type u} [inst : NontriviallyNormedField 𝕜] [inst_1 : NormedAddCommGroup E]
  [inst_2 : NormedSpace 𝕜 E] [inst_3 : NormedAddCommGroup E'] [inst_4 : NormedSpace 𝕜 E']
  [inst_5 : NormedAddCommGroup E''] [inst_6 : NormedSpace 𝕜 E''] [inst_7 : NormedAddCommGroup E''']
  [inst_8 : NormedSpace 𝕜 E'''] [inst_9 : TopologicalSpace H] [inst_10 : TopologicalSpace H']
  [inst_11 : TopologicalSpace G] [inst_12 : TopologicalSpace G'] {I : ModelWithCorners 𝕜 E H}
  {I' : ModelWithCorners 𝕜 E' H'} {J : ModelWithCorners 𝕜 E'' G} {J' : ModelWithCorners 𝕜 E''' G'} {M : Type u_11}
  {M' : Type u_12} {N : Type u_13} {N' : Type u_14} [inst_13 : TopologicalSpace M] [inst_14 : ChartedSpace H M]
  [inst_15 : TopologicalSpace M'] [inst_16 : ChartedSpace H' M'] [inst_17 : TopologicalSpace N]
  [inst_18 : ChartedSpace G N] [inst_19 : TopologicalSpace N'] [inst_20 : ChartedSpace G' N'] {n : WithTop ℕ∞} {x : M}
  {f : M → N} {g : M' → N'} {x' : M'} [IsManifold I n M] [IsManifold I' n M'] [IsManifold J n N] [IsManifold J' n N'],
  Manifold.IsSubmersionAt I J n f x →
    Manifold.IsSubmersionAt I' J' n g x' → Manifold.IsSubmersionAt (I.prod I') (J.prod J') n (Prod.map f g) (x, x')

If f: M → N and g: M' → N' are submersions at x and x', respectively, then f × g: M × M' → N × N' is a submersion at (x, x').

Defined in

Mathlib.Geometry.Manifold.Submersion

Cited by

0 results in Mathlib

Foundations

Depth 210 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NontriviallyNormedFieldNormedAddCommGroupNormedSpaceNormedAddCommGroupNormedSpaceNormedAddCommGroupNormedSpaceNormedAddCommGroupNormedSpaceTopologicalSpaceTopologicalSpaceTopologicalSpaceTopologicalSpaceTopologicalSpaceChartedSpaceTopologicalSpaceChartedSpaceTopologicalSpaceChartedSpaceTopologicalSpaceChartedSpaceIsManifoldIsManifoldIsManifoldIsManifold